1. Field of the Invention
The present invention relates generally to computer vision and imaging systems, and more particularly, to a system and method for segmentation of 2D medical structures by mean shift based ray propagation. More specifically, the present invention relates to segmentation of vessel cross-sections in contrast enhanced CT (computerized tomography) and MR (magnetic resonance) images.
2. Description of the Related Art
Images produced by contrast-enhanced magnetic resonance angiography (CE-MRA) and computed tomography angiography (CTA) are becoming an increasingly important tool in the field of medicine. In the CE-MRA imaging protocol, a contrast agent, usually based on the rare-earth element Gadolinium (Gd) (a highly paramagnetic substance), is injected into the bloodstream. In such images, blood vessels and organs perfused with the contrast agent appear substantially brighter than surrounding tissues. In CTA, a contrast agent is injected into the bloodstream which increases the radio-opacity of the blood making the vessels appear dense.
The goal of the majority of CTA/CE-MRA examinations is diagnosis and qualitative or quantitative assessment of pathologies in the circulatory system. The most common pathologies are aneurysms and stenosis caused by arterial plaques. The modern clinical workflow for the reading of these images increasingly involves interactive 3D visualization methods, such as volume rendering for quickly pinpointing the location of the pathology. Once the location of the pathology is determined, quantitative measurements can be made on the original 2D slice data or, more commonly, on 2D multi-planar reformat (MPR) images produced at user-selected positions and orientations in the volume. In the quantification of stenosis, it is desirable to produce a cross-sectional area/radius profile of a vessel so that one can compare pathological regions to patent (healthy) regions of the same vessel, i.e., to segment the vessel from the background of an image.
There is an extensive body of work on the segmentation of structures (such as air-ways) from CTA and MRA images. This work is based largely on deformable models such as snakes, balloons, levels-sets, and region-competition.
Active contours, or snakes are deformable models based on energy minimization of controlled-continuity splines. When snakes are placed near the boundary of objects to be segmented, they will lock onto salient image features under the guidance of internal and external forces. For example, let C(s)=(x(s),y(s)) be the coordinates of a point on the snake, where s is the length parameter. The energy functional of a snake is defined as                               E          ⁡                      (            C            )                          =                              ∫            0            1                    ⁢                                    [                                                                    E                    int                                    ⁡                                      (                                          C                      ⁡                                              (                        s                        )                                                              )                                                  +                                                      E                    image                                    ⁡                                      (                                          C                      ⁡                                              (                        s                        )                                                              )                                                  +                                                      E                    con                                    ⁡                                      (                                          C                      ⁡                                              (                        s                        )                                                              )                                                              ]                        ⁢                          ⅆ              s                                                          (        1        )            where Eint represents the internal energy of the spline due to bending, Eimage represents image forces, and Econ are the external constraint forces. First, the internal energy,Eint=w1|C′(s)|2+w2|C″(s)|2,  (2)imposes regularity on the curve, where w1 and w2 corresponds to elasticity and rigidity, respectively. Second, the image forces are responsible for pushing the snake towards salient image features. The local behavior of a snake can be studied by considering the Euler-Lagrange equation,                     {                                                                                                                        -                                                                        (                                                                                    w                              1                                                        ⁢                                                          C                              ′                                                                                )                                                ′                                                              +                                                                  (                                                                              w                            2                                                    ⁢                                                      C                            ″                                                                          )                                            ″                                                        =                                      F                    ⁡                                          (                      C                      )                                                                      ,                                                                                                          C                  ⁡                                      (                    0                    )                                                  ,                                                      C                    ′                                    ⁡                                      (                    0                    )                                                  ,                                                      C                    ⁡                                          (                      1                      )                                                        ⁢                                                                           ⁢                  and                  ⁢                                                                           ⁢                                                            C                      ′                                        ⁡                                          (                      1                      )                                                        ⁢                                                                           ⁢                  given                                ,                                                                        (        3        )            where F(C) captures the image and external constraint forces. Note that the energy surface E is typically not convex and can have several local minima. Therefore, to reach the solution closest to the initialized snake, the associated dynamic problem is solved instead of the static problem. When the solution C(t) stabilizes, a solution to the static problem is achieved.                     {                                                                                                                        ∂                      C                                                              ∂                      t                                                        -                                                            (                                                                        w                          1                                                ⁢                                                  C                          ′                                                                    )                                        ′                                    +                                                            (                                                                        w                          2                                                ⁢                                                  C                          ″                                                                    )                                        ″                                                  =                                  F                  ⁡                                      (                    C                    )                                                                                                                                            initial                                +                                  boundary conditions                                                                                        (        4        )            
Snakes perform well when they are placed close to the desired shapes to be segmented. However, a number of fundamental difficulties remain; in particular, snakes heavily rely on a proper initialization close to the boundary of an object of interest and multiple initializations are required when multiple objects are to be segmented, i.e., one per object of interest.
To overcome some of the initialization difficulties with snakes, a balloon model was introduced as a new deformable model based on the snakes idea. This model resembles a “balloon” which is inflated by an additional force which pushes the active contour to object boundaries, even when it is initialized far from the initial boundary. However, the balloon model, like snakes, cannot perform topological changes, i.e., balloons can not merge with other balloons and balloon can not split.
Since snakes and balloons cannot easily capture topological changes, the snake or balloon methods require extensive user interaction for images with multiple objects. This problem can be resolved by the use of the level set evolution, as is known in the art. Under the level set approach, consider a curve C as the zero level set of a surface, φ(x, y)=0. The zero level set of the function φ,{xεR2:φ(t,x)=0}, evolves in the normal direction according to                                                         ∂              ϕ                                      ∂              t                                =                                    g              ⁡                              (                                  x                  ,                  y                                )                                      ⁢                                                        ∇                ϕ                                                    ⁢                          (                                                div                  ⁡                                      (                                                                  ∇                        ϕ                                                                                                                      ∇                                                                                                           ⁢                          ϕ                                                                                                              )                                                  +                v                            )                                      ,                            (        5        )            where             g      ⁢              (                  x          ,          y                )              =          1              1        +                              (                                          ∇                                                                   ⁢                                  G                  σ                                            *              I                        )                    2                      ,v is a positive real constant, Gσ*I is the convolution of the image I with the Gaussian Gσ, and φ0 is the initial data which is a smoothed version of the function 1−XΓ, where XΓ is the characteristic function of a set Γ containing the object of interest in the image. The gradient of the surface ∇φ is the normal to the level set C,{overscore (N)}, and the term div   div  ⁡      (                  ∇        ϕ                                      ∇          ϕ                              )  is its curvature K.
Unlike snakes, this active contour model is intrinsic, stable, i.e., the PDE (partial differential equation) satisfies the maximum principle, and can handle topological changes such as merging and splitting without any computational difficulty. A key disadvantage of the level set method is its high computational complexity, due to the additional embedding dimensional, even when the computation is restricted to a narrow band around the curve. To overcome the computational complexity, narrow band level set evolutions have proposed. However, these methods are still not fast enough for real-time image segmentation.